01-平均差「Mean Deviation」
怎样计算
先求数据的平均值……接着求距离……最后求距离的平均!
三部曲:
- 一、求所有数据值的平均值
- 二、求每个数据与平均值的距离(用数据值减去平均值,不理正负号)
- 三、求这些距离的平均
像这样:
例子:3、6、6、7、8、11、15、16 的 平均差
一、求平均:
平均=
二、求每个数据只与平均值的距离:
值 | 与 9 的距离 |
---|---|
3 | 6 |
6 | 3 |
6 | 3 |
7 | 2 |
8 | 1 |
11 | 2 |
15 | 6 |
16 | 7 |
像这样:
(没有正负号!)
三、求这些距离的平均:
平均差=
所以,平均值 = 9,平均差 = 3.75
平均差告诉我们,每个数据值平均离中心有多远。
在上面的例子里,数据值平均离中心 3.75 远。
在这里,差就是距离
公式
公式是:
平均差 =
我们详细看看:
绝对偏差
在公式里求的每个距离是叫绝对偏差,因为它是偏差的绝对值(离平均值的距离)
我们把 "|" 放在项的两变来代表 "绝对值":
|-3| = 3
为每一个 x:
绝对偏差 = |x - μ|
在这个例子里,16 的 绝对偏差 = |x - μ| = |16 - 9| = |7| = 7
现在把它们加起来……
总和符号
"加起来" 的符号是 Σ(叫总和符号),所以:
绝对偏差的总和 = Σ|x - μ|
除以数据的个数 N,得到:
平均差 =
我们再用正确的符号做一次:
例子: 3、6、6、7、8、11、15、16 的 平均差
一、求平均:
二、求绝对偏差:
三、求平均差:
平均差 =
注意:平均差有时也叫平均绝对偏差(英语 Mean Absolute Deviation (MAD)),因为它是绝对偏差的平均。
平均差的意思是什么?
平均差告诉我们,每个数据值平均离中心有多远。
以下是个例子(用 标准差 网页里的数据):
例子:你和你的朋友测量了你们的狗的身高(毫米):
身高(到肩膀)是:600mm、470mm、170mm、430mm 和 300mm。
一、求平均:
二、求绝对偏差:
x | |x - μ| |
---|---|
600 | 206 |
470 | 76 |
170 | 224 |
430 | 36 |
300 | 94 |
Σ|x - μ| = 636 |
三、求平均差:
平均差=
所以,狗的身高与平均值的平均距离是 127.2 mm。
(可以和 147 mm 的标准差比较一下)
检测
在平均值一边的偏差应该和另一边的偏差是相等的。
上面的例子:
例子:3、6、6、7、8、11、15、16
偏差是:
6 + 3 + 3 + 2 + 1 = 2 + 6 + 7
15 = 15
同样:
例子:狗
平均左边的偏差:224 + 94 = 318
平均右边的偏差:206 + 76 + 36 = 318
如果它们不相同……你便是做错了!
Question
Question 1
Calculate the mean deviation for the numbers: 75, 83, 96, 100, 121 and 125
A. 14.33
B. 15
C. 15.33
D. 16
Q1 Answer
Step 1 Find the mean:
μ = (75 + 83 + 96 + 100 + 121 + 125) ÷ 6 = 600 ÷ 6 = 100
Step 2 Find the Absolute Deviations:
Step 3 Find the Mean Deviation:
Question 2
Ten friends scored the following marks in their end-of-year math exam: 23%, 37%, 45%, 49%, 56%, 63%, 63%, 70%, 72% and 82%
What was the mean deviation of their marks?
A. 14%
B. 15%
C. 16%
D. 17%
Q2 Answer
Step 1 Find the mean:
μ = (23 + 37 + 45 + 49 + 56 + 63 + 63 + 70 + 72 + 82) ÷ 10 = 560 ÷ 10 = 56%
Step 2 Find the Absolute Deviations:
Step 3 Find the Mean Deviation:
Question 3
A booklet has 12 pages with the following numbers of words: 271, 354, 296, 301, 333, 326, 285, 298, 327, 316, 287 and 314
What is the mean deviation of the number of words per page?
A. 18.5
B. 19.33
C. 20
D. 20.17
Q3 Answer
Step 1 Find the mean:
μ = (271 + 354 + 296 + 301 + 333 + 326 + 285 + 298 + 327 + 316 + 287 + 314) ÷ 12
= 3,708 ÷ 12
= 309
Step 2 Find the Absolute Deviations:
Step 3 Find the Mean Deviation:
Question 4
What is the mean deviation of the first 10 numbers of the Fibonacci sequence?
A. 8.8
B. 8.5
C. 8.42
D. 8.32
Q4 Answer
The first 10 numbers of the Fibonacci sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34
Step 1 Find the mean:
μ = (0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34) ÷ 10 = 88 ÷ 10 = 8.8
Step 2 Find the Absolute Deviations:
Step 3 Find the Mean Deviation:
Question 5
The Lakers scored the following numbers of goals in their last twenty matches: 3, 0, 1, 5, 4, 3, 2, 6, 4, 2, 3, 3, 0, 7, 1, 1, 2, 3, 4, 6
Calculate the Mean Deviation.
A. 1.5
B. 2
C. 2.5
D. 3
Q5 Answer
Method 1
Step 1 Find the mean:
μ = (3 + 0 + 1 + 5 + 4 + 3 + 2 + 6 + 4 + 2 + 3 + 3 + 0 + 7 + 1 + 1 + 2 + 3 + 4 + 6) ÷ 20
= 60 ÷ 20
= 3
Step 2 Find the Absolute Deviations:
Step 3 Find the Mean Deviation:
Method 2
This can be done much more neatly using tables, as follows:
First complete a frequency table:
Then calculate the mean and mean deviation from the following table:
Question 6
Ramiro did a survey of the number of pets owned by his classmates, with the following results:
What was the mean deviation of the number of pets?
A. 1.02
B. 1.053
C. 1.087
D. 1.12
Q6 Answer
Calculate the mean and mean deviation from the following table:
Question 7
Emma rolled a die a number of times and recorded her results in a bar graph, as follows:
What was the mean deviation of the scores?
A. 1.375
B. 1.5
C. 1.625
D. 1.875
Q7 Answer
First complete a frequency table:
Then calculate the mean and mean deviation from the following table:
Question 8
Rachel rolled a die forty five times with the following results:
What was the mean deviation of her scores?
A. 1.564
B. 1.609
C. 1.64
D. 1.653
Q8 Answer
Calculate the mean and mean deviation from the following table:
Question 9
A spinner is in the shape of a regular heptagon marked with the numbers 1 to 7.
Theodore spun the spinner 50 times and recorded his results in a table, as follows:
Calculate the mean deviation of the scores.
A. 1.6088
B. 1.7288
C. 1.7688
D. 1.8088
Q9 Answer
Calculate the mean and mean deviation from the following table:
Question 10
The children in a class did a survey of the number of siblings (brothers and sisters) each of them had. The results are recorded in the following table:
Calculate the mean deviation of the number of siblings.
A. 1.38
B. 1.41
C. 1.45
D. 1.48
Q10 Answer
Calculate the mean and mean deviation from the following table:
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